Estimates for Mittag-Leffler functions with smooth phase depending on two variables

Abstract: In this paper we consider the problem on estimates for Mittag-Leffler functions with the smooth phase functions of two variables having singularities of type $D_{\infty} $, $D_{4}^{\pm}$ and $A_{r}$. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals. We extend results of paper \cite{Ruzhansky} and \cite{Ruzhansky2021} to two-dimensional integrals with phase having some simple singularities.